Forecasting link utilization between points of presence in an IP network

ABSTRACT

The present invention provides a method for computing traffic between a pair of Points of Presence in an IP network by summing link utilization values measured for each link connecting a pair of Points of Presence and dividing the sum by the number of link utilization values included in the sum. The resulting average link utilization is the average link utilization of any link in the aggregate and may be multiplied by the number of active links connecting a pair of Points of Presence to reflect the total amount of traffic between the two Points of Presence. Future link utilization may be forecast by modeling the observed traffic between a pair of Points of Presence using wavelet multiresolution analysis to create an approximation curve that captures the long-term trend of link utilization and at least one detail curve that captures the short term deviation of link utilization around the long-term trend. A time series model of the approximation curve may then be constructed and used for forecasting. In a similar fashion, deviation of link utilization may be forecast.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.12/414,272, filed Mar. 30, 2009, entitled “Forecasting Link UtilizationBetween Points Of Presence In An IP Network” which is a continuation ofU.S. application Ser. No. 10/616,673, filed Jul. 10, 2003, entitled“Forecasting Link Utilization Between Points Of Presence In An IPNetwork,” the entirety of each of which is incorporated herein byreference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

None.

TECHNICAL FIELD

The present invention relates to internet protocol networks. Moreparticularly, the present invention relates to the computation ofaggregate traffic between adjacent Points of Presence in an internetprotocol network and the forecasting of future link utilization betweenPoints of Presence in an internet protocol network.

BACKGROUND OF THE INVENTION

Internet protocol networks, often referred to as “IP networks” carrydata throughout the United States and abroad. The data carried on an IPnetwork may be related to internet transmissions, but may also includeother types of transmissions, such as voice transmissions using voiceover IP protocols, or any other type of data formatted for transmissionusing internet protocols.

IP networks typically comprise very high bandwidth optical carriers,such as OC-48 and OC-192 links. These high capacity links connect thePoints of Presence of the network. Each Point of Presence contains oneor more routers in the same geographical location that receive anddirect data packets over the links of the IP network. A single Point ofPresence often referred to as a “PoP”, will often be directly connectedto multiple other PoPs. Any pair of PoPs may be connected by a pluralityof links, typically of equal capacity. An IP network may be highlydynamic, for example due to link changes as links fail, as links areserviced, and as new links are added.

Establishing new links between a pair of Points of Presence, a processreferred to as “provisioning”, often requires a long timeframe, often atleast several months. For this reason, accurately predicting futuredemand on links between PoPs several months into the future is criticalfor capacity planning purposes. If the operator of an IP network doesnot begin the process of provisioning new links before traffic between apair of adjacent PoPs has exceeded the target capacity of the network,the IP network will be compromised for a considerable time while the newlink is established.

BRIEF SUMMARY OF THE INVENTION

The present invention provides a method for computing aggregate IPnetwork traffic between adjacent PoPs and for forecasting future IPnetwork traffic between adjacent PoPs. Adjacent PoPs are PoPs that areconnected directly via a link with no intermediate PoP. Both thecomputation of aggregate traffic and the forecasting of future trafficare useful for IP network capacity planning.

Current IP network usage is measured and computed by measuring linkutilization at predetermined intervals. Further topological informationis obtained to identify the links directly connecting PoPs in the IPnetwork. The total demand between any pair of adjacent Points ofPresence may be computed by summing the utilization values collected forthe links connecting those Points of Presence, while the average demandbetween those Points of Presence may be computed by dividing the sum bythe number of utilization values included in the sum.

Future IP network demand can be forecast, in accordance with the presentinvention, by first modeling collected network utilization data as anapproximation signal. Time series models of the approximation signal maythen be constructed and evaluated in comparison to the collected linkutilization data. The linear time series model that best matchescollected link utilization data can then be used to forecast future linkutilization demands. In a similar fashion, the deviation of linkutilization may be calculated and forecast.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

The present invention is described in detail below with reference to theattached drawing figures, wherein:

FIG. 1 schematically illustrates a portion of an IP network, the trafficof which may be computed and forecast in accordance with the presentinvention;

FIG. 2 illustrates a method in accordance with the present invention forcollecting link utilization data and computing aggregate demand;

FIG. 3 illustrates a method in accordance with the present invention forcalculating average link demand;

FIG. 4 illustrates a method in accordance with the present invention forforecasting future link utilization;

FIG. 5 illustrates a method in accordance with the present invention forforecasting future deviation of link utilization;

FIG. 6 illustrates collected link utilization data;

FIG. 7 illustrates a collected link utilization data at a small timescale;

FIG. 8 illustrates the periodicity of collected link utilization data inthe form of Fourier transforms of collected link utilization data;

FIG. 9 illustrates an á trous wavelet transform that may be used inaccordance with the present invention to model collected linkutilization data;

FIG. 10 illustrates approximation signals modeling the collected linkutilization data;

FIG. 11 illustrates detail signals used in modeling the collected linkutilization data;

FIG. 12 illustrates the energy distribution for the detail signalsillustrated in FIG. 11;

FIG. 13 illustrates the link utilization approximation signal of onelink and the average daily standard deviation for the link utilization;

FIG. 14 illustrates an approximation of collected link utilization datausing the average weekly long-term trend and the average daily standarddeviation;

FIG. 15 illustrates a forecast of link utilization made in accordancewith the present invention;

FIG. 16 illustrates a weekly relative prediction error for a linkutilization forecast made in accordance with the present invention;

FIG. 17 illustrates weekly link utilization forecasts for a link made inaccordance with the present invention; and

FIG. 18 illustrates adjusted weekly link utilization forecasts for alink made in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 schematically illustrates a portion 100 of an IP network. Thepresent invention may be utilized in an IP network such as illustratedby portion 100. Network portion 100 includes a plurality of Points ofPresence, such as PoP 110. Each PoP is connected to at least one otherPoP by a link. For example, the first PoP 110 connects to a second PoP120 by link 116 and link 117. The first PoP connects to a third PoP 130by link 111 and link 112. The first PoP 110 may also connect to a fourthPoP 140 by link 113, link 114, and link 115. The second PoP 120 mayconnect to the third PoP 130 by link 121, link 122, and link 123. Thesecond PoP may connect to the fourth PoP 140 by link 124 and link 125.The third PoP 130 may connect to the fourth PoP 140 by link 131 and link132. The fourth PoP 140 may connect to a fifth PoP 150 by link 146 andlink 147. The fourth PoP 140 may connect to a sixth PoP 160 by link 141and link 142. The fourth PoP 140 may connect to a seventh PoP 170 bylink 143, link 144 and link 145. The fifth PoP 150 may connect to thesixth PoP 160 by link 151, link 152, and link 153. The fifth PoP 150 mayconnect to the seventh PoP by link 154 and link 155. The sixth PoP 160may connect to the seventh PoP 170 by link 161 and link 162. It shouldbe realized that the portion of an IP network 100 illustrated in FIG. 1is illustrative only. In actuality, an IP network may contain adifferent number of PoPs than those illustrated in FIG. 1, oftenconsiderably more than the seven illustrated, and may likewise contain adifferent number of links than is illustrated in FIG. 1, for exampleconsiderably more. Furthermore, any linked pair of PoPs may be connectedby any number of links, rather than the two or three links illustratedin FIG. 1. While a single link may be used to connect a pair of PoPs, alarge plurality of links may also be used. The number of links used toconnect a pair of Points of Presence is a network design decision thatmay be aided by the practice of the present invention.

The Points of Presence illustrated in FIG. 1, are well known in the artand may comprise any number of physical structures allowing a serviceprovider to provide services at a given location. Typically, a PoP mayinclude a router or similar device to receive and direct data packetsover the links.

Link utilization data may be collected from operational routers using,for example, the Simple Network Management Protocol (SNMP). Routers maycount the number of packets or bytes transmitted over a particular IPlink and report those measurements using SNMP upon request. A networkelement, usually referred to as a Network Management Station (NMS), maybe configured to initiate SNMP requests to operational routers in the IPnetwork at predetermined intervals to collect available link utilizationdata. Because a single network element may not be able to poll theentire network simultaneously, if a single NMS is used the NMS may pollthe entire network within a given period of time, such as five minutes,meaning that the SNMP collection may be asynchronous. Furthermore, theSNMP protocol is an unreliable IP protocol wherein packets may bedropped in a connection without specific notification to thecommunicating entities. As a result, there may be cases when NMS hasissued a request to a router but that router does not reply, due to thereply being lost in the network or due to the router being unable toreply because of other resource intensive tasks. The use of SNMP tocollect link utilization data requires that the problems of asynchronousdata collection and missing data be overcome. The failure to account forthese problems may prevent link utilization models and forecasts fromaccurately reflecting actual IP network behavior.

The asynchronous data collection may be accounted for by measuring linkutilization over a time period greater than the polling interval. Forexample, for a five minute polling interval, link utilization may bemeasured over a ninety minute interval. The duration of the time periodof the link utilization measurement relative to the duration of thepolling interval may vary greatly, although generally the time period ofthe link utilization measurement may be at least twice the duration ofthe polling period. This allows all link utilization measurements afterthe beginning and before the end of the time period to be used even ifthe data's arrival at the NMS is delayed.

However, the problem of lost data remains. The problem of missing datarelates to two different types of missing data. First, some, but notall, utilization values may be missing for a given link. Second, alllink utilization values may be missing for a link in the aggregate oflinks connecting a pair of PoPs.

To overcome the problem of some missing utilization values for a givenlink, the average aggregate demand between two PoPs may be computed asthe sum of the average utilization of each link. The utilizationmeasurements for a link may be summed, and the sum then divided by thenumber of utilization measurements included in the sum. This process maybe completed for each link in the aggregate, with the results for eachlink then used to compute an average across all links in the aggregate.Accordingly, for a pair of PoPs having N links between them, with acomputed average link utilization 1, for each of the N links, theaverage link utilization between the pair of PoPs may be calculated as:

$( {\sum\limits_{i = 1}^{N}l_{i}} )/N$

This technique will provide an accurate measure for the averageaggregate demand when the missing values result in at least onemeasurement for each link.

However, if all measurements for a link in one time interval are lost,this methodology will lead to an inaccurate computation of the averageaggregate demand, since a zero link utilization will be included in theaverage for the link with the missing values, even though the link mayhave been active. To counteract the effect of missing data for an entiretime period for a link, the method for measuring aggregate demand may bemodified. For each time interval, the SNMP measurements for each link inthe aggregate is collected. The collected link utilization values aresummed and then divided by the number of values included in the sum. Forexample, a pair of PoPs may be connected by an aggregate of five links.If, by way of further example, link utilization values are collected atfive minute polling periods and link utilization is to be measured overa ten minute time period, then if there is no data loss the total numberof link utilization values should be ten, which is two values per linkfor five links. In this example, the average aggregate link utilizationwould be computed by dividing the sum by ten. In actual practice, somelink utilization values for a single link may be missing for the entiremeasurement time period. By summing link utilization for all links inthe aggregate and dividing by the number of measurements, a moreaccurate result is obtained for the frequent case where all linksconnecting a pair of PoPs have equal capacity. Multiplication of theaverage aggregate link utilization by the number of links in theaggregate may be used to provide a measure of the total traffic betweena PoP pair.

Referring now to FIG. 2, a method 200 for determining aggregate demandbetween a pair of adjacent Points of Presence in accordance with thepresent invention is illustrated. In step 210 utilization values forlinks are collected. While step 210 may involve collecting linkutilization information for all links in an IP network, as explainedabove, the collection methodology may not be error-proof, therebyresulting in utilization values not being collected for every link inthe IP network for every collection period. Step 210 may involve therouters in the PoPs of an IP network reporting link utilization values.As further described herein, step 210 may be implemented over apredetermined polling period. The utilization values collected in step210 may be in a variety of forms, such as the total number of bytesreceived or transmitted, an exponentially weighted moving average, orany other measurement of link utilization. Whatever form is used forlink utilization values, link utilization may be reported for a firstshort time frame. The first short time frame may be as little as a fewsecond, or even less, to several minutes. As explained below, linkutilization values for the first short time frame may be averaged over asecond longer time frame. The first short time frame may correspond tothe polling period, but need not be identical to the polling period. Instep 220, topological information is gathered for all links. Thetopological information gathered in step 220 identifies which link orlinks connect each pair of adjacent PoPs. Step 220 may be performed atpredetermined time intervals. The predetermined time intervals at whichstep 220 is performed may vary greatly, from the very frequent for arapidly changing network to the highly infrequent for a very staticnetwork. A time interval of one week may be used for many IP networks.The topological information collected in step 220 may be correlated withthe link utilization information collected in step 220 so that theutilization of a specific link may be known.

Referring now to FIG. 3, a method 300 for computing aggregate demand isillustrated. In step 310, the source and destination PoPs are identifiedfor all links with collected link utilization data. This may be done bycorrelating the collected topological information with the collectedlink utilization values. As a result of step 310, it is known whichlinks connect which Points of Presence and the utilization of thoselinks. In step 320, link utilization values are summed for all linksbetween a source PoP and a destination PoP for a given time period, thetime period being a second longer time frame as compared to the firstshort time frame over which the link utilization values were measured.The time period used in step 320 may be at least twice the duration ofthe polling period of step 210 of method 200 if method 300 uses datacollected using method 200. In step 330, the average aggregate demand iscalculated by dividing the sum of step 320 by the number of linkutilization values included in that sum. The aggregate link utilizationbetween the adjacent PoPs may be computed by multiplying the average bythe number of active links between the pair of PoPs.

For example, to compute the aggregate demand between Point of Presence110 and Point of Presence 140 illustrated in FIG. 1 link utilizationvalues would be collected for link 113, link 114, and link 115.Collected topological information would identify these links, namelylink 113, link 114, and link 115, as the links connecting PoP 110 andPoP 140. The link utilization values for link 113, link 114, and link115 for the relevant time period would be summed. This sum of linkutilization would be divided by the number of collected measurements forall three links in the aggregate connecting Point of Presence 110 andPoint of Presence 140.

It should be realized that FIG. 2 and FIG. 3 broadly illustrate methodsin accordance with the present invention to collect link utilizationinformation and to compute aggregate demand. It should be appreciatedthat the steps of the method 200 and method 300 may be performed invarying orders or may occur simultaneously. It should also be realizedthat some steps, such as collecting link utilization data in step 210and collecting topological information in step 310, may be omitted, forexample if the information to be collected is already available. Oneexample of an application of the present invention is described below.It should be realized that applications of the method described anddisclosed in FIG. 2 and FIG. 3 may be employed in ways beyond theexamples described below. For example, in the management of large-scaleIP networks it may be more convenient to picture an IP network at thegranularity of PoPs and aggregate pipes between PoPs, rather thanrouters and IP links. Further network design principles may be readilydeployed on networks abstracted at the PoP level, where IP linkutilization is substituted by the average amount of traffic flowingbetween adjacent PoPs over the multiplicity of links interconnecting theadjacent PoPs.

Referring now to FIG. 4, a method 400 for forecasting future linkutilization is illustrated. In step 410, collected link utilization datais modeled as a link utilization approximation signal. One example ofappropriately modeling collected link utilization data is the use ofwavelet multiresolution analysis, including á-trous modeling techniquesas discussed below, although other modeling methods may be used inconjunction with the present invention. In step 420, a time series modelis constructed for the link utilization approximation signal. The timeseries model constructed in step 420 may be a linear model, therebysimplifying the construction of the model. Examples of the constructionof appropriate linear time series models of the link utilizationapproximation signal are described below, such as the use ofauto-regressive modeling, moving average modeling, and auto-regressivemoving average modeling. As described further below, an ARIMA model maybe constructed in step 420. It should be understood, however, that othermodels of the link utilization approximation signal may be used. A largenumber of models may be constructed in conjunction with step 420. Instep 430, the quality of the time series models' description of the linkutilization approximation signal is determined. Step 430 determineswhich of the models best describes the link utilization approximationsignal. One skilled in the art will realize that a variety of criteriamay be used to determine which model is the best fit for the data. Avariety of methods may be used in step 430, such as those described inthe example below. In step 440, future link utilization is forecastusing the time series model that best matches the collected linkutilization data. It should be noted that if only one time series modelis constructed, step 430 of determining the quality of the time seriesmodel may be omitted, and in step 440 future link utilization will beforecast using that time series model. However, the use of a single timeseries model will run a high risk of poorly fitting collected linkutilization data and providing a correspondingly poor forecast of futurelink utilization. Accordingly, while the use of a single time seriesmodel is within the scope of the present invention, the use of aplurality of time series models may often provide a better forecast, dueto there being a larger number of models from which to choose the modelthat best fits the approximation signal.

Referring now to FIG. 5, a method 500 for forecasting future deviationaround the forecast link utilization baseline is illustrated. Thedeviation of link utilization may be thought of as the fluctuation oflink utilization around the long-term trend. As explained below in anexemplary description of a method in accordance with the presentinvention, link utilization over time may demonstrate both a long-termtrend and short term periodicities. The short term periodicities may be,for example, over periods of twelve hours, twenty-four hours, and aweek. In modeling and forecasting link utilization, accounting for suchdeviation may be important to accurately plan future network capacity.Because the deviation occurs around the long-term trend, even aperfectly accurate forecast of the long-term trend of the utilization ofa link may overstate or, more problematically for network capacityplanning, understate link demand, due to this deviation. Accordingly,method 500 for forecasting future link utilization deviation may be usedto account for this deviation of link utilization.

In step 510, the deviation of collected link utilization data is modeledas a deviation approximation signal. It should be noted that thedeviation approximation signal constructed in step 510 of method 500 maybe constructed in conjunction with the link utilization approximationsignal constructed in step 410 of method 400. As described in theexample of one use of methods in accordance with the invention describedbelow, the deviation approximation signal may be selected from thedetail signals constructed using wavelet multi-resolution analysisand/or á-trous modeling to reconstruct the collected link utilizationdata. It should be appreciated that other modeling techniques may beused, however. In step 520, a time series model of the deviationapproximation signal is constructed. The time series model constructedin step 520 may be a linear model, thereby simplifying the constructionof the model. One example of linear time series models is describedbelow, such as the use of to auto-regressive modeling, moving averagemodeling, and auto-regressive moving average modeling. Of course, othermodeling methods may be used. In step 530, the quality of each timeseries model of the deviation approximation signal is evaluated ascompared to the computed deviation approximation signal. The examplebelow describes some methods for evaluating the time series models,although other methods may be used. In step 540, future link utilizationdeviation is forecast using the time series model that best matches thedeviation of the collected link utilization data. It should be notedthat if only one time series model is constructed, step 530 ofdetermining the quality of the time series model may be omitted, and instep 540 future deviation of link utilization will be forecast usingthat time series model. As with method 400, however, the use of a singletime series model of link utilization deviation, while within the scopeof the present invention, may not provide an accurate forecast of futurelink utilization deviation due to the possible lack of a model that fitsthe deviation approximation signal well.

A specific example of collecting link utilization data and computingaggregate demand using method 200 and method 300, and then forecasting alink utilization using method 400 and method 500 is described below. Theexample discussed below involves the experimental collection, analysis,modeling, and forecast using link utilization data collected from anexisting IP network. The example described below is an exemplarydescription of one application of the present invention, and is notintended to limit the scope of the present invention, but to merelydemonstrate one possible use of it.

Referring now to FIG. 6, aggregate link utilization obtained usingmethod 200 and method 300 for three traces is shown. A trace representsthe connection between a pair of PoPs, which may include any number oflinks in the aggregate. As can be seen in reference to FIG. 6, the datawas collected from October 2000 to July 2002. Trace utilization is shownas aggregate demand measured in megabits per second. The dataillustrated in FIG. 6 is SNMP data collected from routers for incomingand outgoing link utilization. The collection of the data used todetermine the aggregate values illustrated in FIG. 6 was notsynchronized, in that not all links were polled at the same time.Polling all links simultaneously could overload the IP network and thecollection station. Values collected correspond to an exponentiallyweighted moving average computed on ten second link utilizationmeasurements. The exponential weighted average collected had an averageage of five minutes, with more recent samples being weighted moreheavily than earlier taken samples. These measurements were taken usingobjects in two proprietary management information base objects, whichcollected the data used in a proprietary method not available to theinventors. Other collection methods, including the use of other timeperiods and other averaging methods, may be employed with the presentinvention. Of course, whatever collection method is employed shouldpreferably be consistent for all links. Topological information was alsocollected from the routers. The topological information included therouters identity, the links connected to that router, and thedestination routers for those links. The collected SNMP data wascorrelated with collected topological information so that the linkutilization data could be correlated to the link to which that datarelates. The PoPs containing the routers involved were also identified.

Calculating the aggregate demand illustrated in FIG. 6 requires that,for each link in the SNMP data, its source and destination Point ofPresence be identified. The notation l_(sd)(k) may denote the k^(th)link connecting Point of Presence s to Point of Presence d. Next, timewas discretized into ninety minute intervals. While other time intervalsmay be used, the ninety minute interval was useful for demonstratingperiodicities in the data and facilitating modeling, as shall bediscussed below. Each ninety minute time interval was indicated with anindex t. The aggregate demand for any Point of Presence pair, forexample PoP s and PoP d at a time interval t was calculated as the sumof all the records obtained at that time interval t for all linksbetween that PoP s and PoP d divided by the total number of linkutilization records. The result was the average aggregate demand of alink between PoP s and PoP d at the time interval t. The above-describedapproach allowed for missing values for particular links in theaggregate to be accommodated. Moreover, possible inaccuracies in theSNMP measurements were smoothed by the averaging operation. FIG. 6illustrates this aggregate demand as the time index t advances.

In generally reviewing FIG. 6, which illustrates aggregate linkutilization for three exemplary traces, certain observations can be madeas to the link utilization behavior. Each trace also shows a deviationof link utilization, some greater than others, and sometimes thedeviation changes over time. For example, link utilization for trace oneshows increasing deviation as t advances. Each trace has a differentoverall long-term trend. For example, trace one and trace five showincreasing link utilization as a long-term trend, while trace eight doesnot show an immediately discernible long-term trend of increased linkutilization. There are also several utilization spikes for each trace.These spikes illustrate sharp short term increases in link demand,significant enough in magnitude, duration, or both that they were notsmoothed away by the averaging operation. Such sharp short termincreases in link utilization may result from link failures elsewhere inthe IP network that cause traffic to be rerouted to that link, denial ofservice attacks, routing changes, or other short lived but substantialcircumstances that cause a peak in traffic. While the methods inaccordance with the present invention take this data into account inconstructing models of link utilization and deviation of linkutilization, the methods in accordance with the present invention do notattempt to predict the occurrences of such spikes, thus treating them asoutliers.

Referring now to FIG. 7, a more detailed view of aggregate linkutilization data for a shorter time frame is illustrated. FIG. 7illustrates link utilization for the month of May 2002 for the tracesillustrated in FIG. 6. One observing FIG. 7 will note that there appearto be strong periodicities in the data, mostly on scales of roughly aweek and/or of a day. Not all three traces illustrated show the sameperiodicities or the same degree of periodicity.

FIG. 8 further illustrates the periodicities of the data in FIG. 7discussed above. FIG. 8 illustrates a Fast Fourier Transform of the dataillustrated in FIG. 7 for aggregate link utilization data for the monthof May 2002. As shown in FIG. 8, all three traces exhibit strong periodsof twenty-four hours. Trace one and trace three exhibit a weaker twelvehour periodicity. Trace one and trace two exhibit a weaker, but stillnoticeable, periodicity at one hundred sixty-eight hours, whichcorresponds to one week.

In further reference to FIG. 6, FIG. 7 and FIG. 8, additionalobservations regarding the collected link utilization data may be made.First, utilization of different traces vary in different ways and atdifferent time scales, which is to say there is a multi-time scalevariability across all traces. Second, there are strong periodicities inthe data, although the strength and nature of those periodicities arenot identical for all traces. Third, the collected link utilization datademonstrates evident long-term trends that vary for different traces,which may be described as nonstationary behavior. These generalproperties may be exploited in accordance with the present invention toforecast future behavior.

The collected link utilization data may be modeled using waveletmultiresolution analysis. Wavelet multiresolution analysis describes theprocess of synthesizing a discrete signal, such as the aggregate linkdemand, by beginning with a low resolution signal, i.e., a signal at acoarse time scale, and successively adding details onto that signal tocreate a higher resolution version of the signal. The waveletmultiresolution analysis process ends with a complete synthesis of theoriginal signal at the finest resolution, which is say at the finesttime scale. The finest time scale as described in this example is aninety minute time scale, as the measurements are averaged over aninety-minute period. At each time scale 2^(j), the signal is decomposedinto an approximate signal, or an approximation, and a detailed signalthrough a series of scaling of functions φ(t) and wavelet functions ψ(t)where κεZ is a time index at scale j. The scaling and wavelet functionsare obtained by dilating and translating the mother scaling functionsφ(t), φ_(j,k)(t)=2^(−j/2)φ(2^(−j)t−k) and the mother wavelet functionψ(t), ψ_(j,k)(t)=2^(−j/2)ψ(2^(−j)t−k). The approximation is representedby a series of scaling coefficients a_(j,k) and the detail isrepresented by a series of wavelet coefficients d_(j,k). For a signal,such as the illustrated trace utilization data, denoted x(t), with Ndata points at the finest of time scale, the multiresolution analysiscan be written as:

${x(t)} = {{\sum\limits_{\kappa \in Z}{a_{p,k}{\varphi_{p,k}(t)}}} + {\sum\limits_{0 \leq j \leq p}{\sum\limits_{\kappa \in Z}{d_{j,k}{\psi_{j,k}(t)}}}}}$

In the above equation, p≦log N. The sum of the coefficients a_(p,k)represents the approximation at the coarsest time scale 2^(p), while thesums of coefficients d_(j,k) represent the details on all the scalesbetween 0 and p.

One skilled in the art of signal processing will appreciate that theroles of the mother scaling and wavelet function φ(t) and ψ(t) can bedescribed and represented using a low-pass filter h and a high-passfilter g. As a result, the multiresolution analysis and synthesis of asignal x(t), such as the illustrated trace utilization data, can beimplemented efficiently as a filter bank. The approximation at scale j,{a_(j+l,k)} is passed through the low-pass filter h and the high-passfilter g to produce the approximation, {a_(j+l,k)}, and the detail{d_(j+l,k)}, at scale j+1. At each stage the number of coefficients atscale j is decimated into half at scale j+1, due to down-sampling. Thisdecimation reduces the number of data points to be processed at thecoarser time scales, but can also leave artifacts in the coarser timescale approximations.

A so-called á-trous wavelet transform has also been developed, whichproduces a smoother approximation by filling in the gaps caused by thedecimation, as described above, by using redundant information from theoriginal signal. Under the á-trous wavelet transform, approximations fora signal x(t), such as the illustrated trace utilization data, aredefined at different time scales as:c ₀(t)=x(t)

${c_{j}(t)} = {\sum\limits_{l = {- \infty}}^{\infty}\;{{h(l)}{{c_{j - 1}( {t + {2^{j - 1}l}} )}.}}}$

In the above, 1≦j≦p, and h is a low-pass filter with a compact support.The detail of signal x(t) at scale j is given by:d _(j)(t)=c _(j−1)(t)−c _(j)(t)

If d_(j)={d_(j)(t),1≦t<N} denotes the wavelet coefficient at scale j,and c_(p)={c_(p)(t),1≦t<N} denotes the signal at the lowest resolution,which is often referred to as the residual, then the set of d₁, d₂, . .. , d_(p), c_(p), represents the wavelet transform of the signal up tothe resolution p, and the signal x(t) can be expressed as an expansionof its wavelet coefficients:

${x(t)} = {{c_{p}(t)} + {\sum\limits_{j = l}^{p}{d_{j}(t)}}}$

At this point, the collection of data across ninety minute intervals isparticularly useful. As discussed above with regard to the Fast FourierTransforms of link utilization data and the periodicity of the collectedlink utilization data, the collected measurements exhibit strongperiodicities at the cycles of twelve and twenty-four hours. Usingninety minutes, or one and a half hours, as the finest time scale allowsthe behavior of the time series to be easily examined at the periods ofinterest by observing its behavior at the third time scale (twelvehours) and fourth time scale (twenty-four hours).

To smooth the data using the á-trous wavelet transform, the low passfilter h from Equation 3 may be selected using the B₃ spline filter,defined by ( 1/16, ¼, ⅜, ¼, 1/16). This selection is of compact support,which is necessary for a wavelet transform, and is point symmetric,which prevents the wavelets from experiencing phase shifts and driftingrelative to the original signal. Each level of resolution the B₃ splinefilter gives a signal which is smoother than the one at the previouslevel without distorting possible periodicities in the data and whilepreserving the original structure of the signal.

In order to understand how c_(j)(t) was computed at each time scale j,the computation is schematically represented in FIG. 9. Element c₁(5) iscomputed based on the value c₀(t)=x(t) at times (5−2), (5−1), 5, (5+1),and (5+2). Then c₂(5) may be calculated based on c₁(1), c₁(3), c₁(5),c₁(7), and c₁(9). Moving toward coarser levels of resolution requiresvalues from the previous resolution level which are farther apart fromeach other. It should be noted that c_(p), is defined for each t=1, 2, .. . , n, where n corresponds to the ninety minute, or one and a halfhour, interval at which measurements were taken and is limited by thesize N of the original signal. Accordingly, computing c_(p)(n) requiresvalues c_(p-1) until time n+2^(p), which iteratively requires values ofc_(p-2) until time n+2^(p-1), etc. As a consequence, the calculation ofc_(p)(n) requires that the original time series x(t) have n+Σ_(j=1)^(j=p)2^(j) values. Because the original signal contained N values, thewavelet coefficients up to the sixth resolution level will contain nvalues, where n+Σ_(j=1) ^(j=p)2^(j)=N, or n=N−126.

The results after the analysis as described above are furtherillustrated in FIG. 10 and FIG. 11. FIG. 10 and FIG. 11 show theapproximation and detail signals for trace five at each time scale, thecoarsest of which is the resolution level 2⁶, or ninety-six hours. Thesixth time scale is chosen as the coarsest because it provides asufficiently smooth approximation signal and captures the evolution ofthe time series from week to week without being affected by thefluctuations at twelve and twenty-four hours. FIG. 11 shows the detailsignals for trace five at each of the time scales.

Using the derived decomposition signals, the energy apportioned to theoverall trend in trace c₆ and each one of the detail signals illustratedin FIG. 11 can be calculated. The energy of a signal y(t), where 1≦t≦N,is defined as E=Σ_(t=1) ^(n)y²(t). The results of the energy calculationfor exemplary traces are:

Trace ID 1 2 3 4 c₆ 96.07% 97.20%  95.5%  96.5% c₆ + d₃ 98.10% 98.76%97.93% 97.91% Trace ID 5 6 7 8 c₆ 95.12% 95.99% 95.84% 97.30% c₆ + d₃97.54% 97.60% 97.68% 98.45%

The overall trend c₆ accounts for approximately 95% to 97% of the totalenergy of the signal. If the overall trend is subtracted from the data,a substantial difference in the amount of energy distributed among thedetail signals may be observed.

This difference in energy in the detail signals is illustrated in FIG.12. As FIG. 12 shows, once the overall trend is subtracted from thedata, the maximum amount of energy in the details is located at thethird time scale, which corresponds to periodicity over twelve hours. Ifthe original signal is approximated using the long-term trend, denotedas c₆, and the fluctuations at the twelve hour time scale, which isdenoted d₃, this combination is capable of accounting for 97 to 99% ofthe total energy of the collected link utilization data.

As explained above, the original signal of the collected linkutilization data can be completely reconstructed using the approximationsignal at the sixth time scale and the six detail signals at lower timescales. The wavelet multiresolution approximation described above canalso be conceived of as a multiple linear regression model, where theoriginal signal x(t) is expressed in terms of its coefficients. In sucha multiple linear regression model, the analysis of variance techniqueis a statistical method used to quantify the amount of variabilityaccounted for by each term in a multiple linear regression model. Theanalysis of variance technique can be used in the process of reducing amultiple linear regression model by identifying those terms in theoriginal model that explain the most significant amount of variance.Using the analysis of variance methodology, the amount of variance inthe original signal explained by the sixth approximation signal and eachof the detail signals can be calculated. The results indicate that thedetail signals d₁, d₂, d₅, and d₆, contribute less than 5% each in thevariance of the original signal.

The modeling and forecast of aggregate demand can be facilitated if themodel of the data can be reduced to a simple model of two parameters,one corresponding to the overall long-term trend and the secondaccounting for the bulk of the variability. The overall trend may bedepicted by c₆, and the detail signal d₃, carries the majority of theenergy among all the detail signals. Thus one possible reduced model is:x(t)=c ₆(t)+βd ₃(t)+e(t)

Using the least squares method, the value of β for each of the traceswith collected link utilization data modeled as described above may becalculated. The β calculated for all traces were between 2.1 and 2.3.Using the analysis of variance technique, this model of the signal maybe evaluated with respect to the proportion of variance it accounts for.If x(t) is the collected link utilization data, and if e(t) is the errorincurred in the new model, error may be calculated by definingSSX=Σ_(t=1) ^(n)(x(t))² and SSE=Σ_(t=1) ^(n)=e(t)². The total sum of thesquares, designated SST, may be defined as the uncertainty that would bepresent if one had to predict individual responses without any otherinformation. Thus SSX=Σ_(t=1) ^(n)(x(t)−x)². The analysis of variancemethodology partitions this variability into two parts. One part isaccounted for by the new model. It corresponds to the reduction inuncertainty that occurs as the regression model is used to predict theresponse. The remaining portion is the uncertainty that remains evenafter the model is used. SSR may be defined as the difference betweenSST and SSE. This difference represents the sum of the squares explainedby the regression. The fraction of the variance that is explained by theregression, or SSR/SST determines the quality of the regression and iscalled the coefficient of determination, and is denoted R². The modelmay be considered to be statistically significant if it can account fora large fraction of the variability in the response, that is to say ifit yields large values for R². The results obtained for the value of βand R² for all eight traces examined herein are:

Trace ID 1 2 3 4 β 2.09 2.06 2.11 2.23 R² 0.87 0.94 0.89 0.87 Trace ID 56 7 8 β 2.12 2.18 2.13 2.16 R² 0.92 0.80 0.86 0.91

The new model, as can be seen, explains 80% to 94% of the variance inthe signal.

From the above described wavelet multiresolution analysis severalconclusions may be drawn. First, there is a clear overall long-termtrend present in the traces. Second, the fluctuations around thelong-term trend are mostly due to periodic changes in traffic bandwidthat a time scale of twelve hours. Third, the long-term trend and thedetail signal at the third time scale, representing the twelve hourfluctuation, account for approximately 98% of the total energy of thecollected link utilization data.

From the analysis of variance procedure, further conclusions may bereached. First, the largest amount of variance in the original signalcan be explained by its long-term trend, denoted as c₆, and the detailsignals d₃ and d₄ at the time scales of twelve and twenty-four hours,respectively. Second, the original signal can be sufficientlyapproximated by the long-term trend and its third detail signal, d₃, asa model that explains approximately 90% of the variance in the collectedlink utilization data.

Based upon the above observations, a generic model for the time seriesmay be created. This model is based upon the model above, where β=3, fora common model across the entire backbone. This model may slightlyoverestimate aggregate demand between two adjacent PoPs, but such anoverestimation may be beneficial for capacity planning purposes:χ′(t)=c ₆(t)+3d ₃(t)

For forecasting future link utilization at a time scale of weeks ormonths the short term fluctuations likely need not be accuratelymodeled. Particularly for capacity planning purposes, the IP networkoperator need only know the traffic baseline for the future and thelikely fluctuation of traffic around this baseline. In the equationabove, d₃(t) is defined for every ninety minute interval in themeasurements capturing the short term fluctuations at the time scale oftwelve hours. Because the specific behavior within a day is likely notimportant for capacity planning purposes weeks or months in the future,the standard deviation of d₃ may be calculated for each day. The weeklystandard deviation, denoted dt₃(j), as the average of the sevendeviation values computed within each week. This value represents thefluctuations of traffic around the long-term trend forecast from day today within each particular week.

Referring now to FIG. 13, the aggregate demand for trace five is shownas determined from the collected SNMP data. FIG. 13 further illustratesthe long-term trend in the data, along with two curves showing theapproximation of the signal as the sum of the long-term trend plus orminus three times the average daily standard deviation within a week, asdescribed above. As seen in FIG. 13, approximating the original signalin this manner expresses the fluctuations of the data around thebaseline long-term trend with considerable accuracy. It should be notedthat the new signal, dt₃, features one value every week, expressing theaverage daily standard deviation within that week. The long-term trendmay likewise be approximated with a more compact time series featuringone value for every week. It should be recalled in this regard thatforecasting is likely to be performed weeks or months in advance forcapacity planning purposes, rendering fluctuations over a twenty-fourhour time period unlikely to be important for such forecasting, so longas the deviation of short term periodicities are accounted for in thetotal forecast demand. Given that the sixth approximation signal is avery smooth approximation of the original signal, its average may becalculated across each week and denoted as a new time series l(j)expressing the long-term trend from one week to the next. Thus, theforecasting process will have to predict the behavior of:{circumflex over (x)}(j)=l(j)+3dt ₃(j)where j denotes the index of each week in the trace.

Referring now to FIG. 14, the signal resulting from the above equationis illustrated. As shown, the approximation of the original signal usingweekly average values for the overall long-term trend and the dailystandard deviation results in a model that accurately reflects theobserved behavior. While the forecast link utilization is made on aweekly basis, that weekly forecast incorporates both the long-term trendand the deviation around the long-term trend due to short periodicitiesAccordingly, the use of linear time series models are likely to beuseful in deriving forecasts for future link utilization and deviationaround the forecast future link utilization. Both of these values areuseful for capacity planning purposes weeks or months in advance.

Constructing a time series model for forecasting purposes impliesexpressing X_(t) in terms of previous observations X_(t-j) and noiseterms Z_(t) that correspond to external events. The noise terms Z_(t)may be assumed to be uncorrelated with a zero mean and finite variance.These are the simplest processes to model and are said to have nomemory, because their value at a time t is uncorrelated with all pastvalues up to time t−1.

Several forecasting models are known, including linear models. Threeknown linear forecasting models are the autoregressive model, the movingaverage model, and the autoregressive moving average model. A timeseries x_(t) is an autoregressive moving average model if x_(t) isstationary and if for every t X_(t)−φ₁X_(t-1)− . . .−φ_(p)X_(t-p)=Z_(t)+θ₁Z_(t-1)+ . . . +θ_(q)Z_(t-q). If p=0, then themodel reduces to a pure moving average process, while if q=0 the processreduces to a pure autoregressive process. This equation can also bewritten in a more concise form as:ω(B)X _(t)=θ(B)Z _(t)

In the above equation, φ(•) and θ(•) are the p^(th) and q^(th) degreepolynomials and B is the backward shift operator B^(j)X^(t)=X_(t-j) andB^(j)Z_(t)=Z_(t-j) where j=0, ±1, . . . .

It should be noted that the autoregressive moving average model fittingprocedure assumes that the data is stationary. The data may not bestationary in many applications of the present invention. If a timeseries exhibits variations that render it nonstationary, there areapproaches known in the art that may be used to render the time seriesstationary. One such method is what is often referred to as adifferencing operation, where the lag-1 difference operator ∇ may bedefined by:∇X _(t) =X _(t) −X _(t-1)=(1−B)X _(t)

In the above equation, B is the backward shift operator. If thenonstationary portion of a time series is a polynomial function of time,then differencing a finite number of times can reduce the time series toan autoregressive moving average process. An ARIMA (p, d, q) model, isan autoregressive moving average model that has been differenced dtimes. It may be written in the form:ω(B)(1−B)^(d) X _(t)=θ(B)Z _(t) ,Z _(t) ˜WN(0,σ²)

If the time series has a non-zero average value through time, the aboveequation also includes a constant term μ on the right hand side.

The above techniques have been verified by use upon collected linkutilization data. In order to model the components l(j) and dt₃(j) usinglinear time series models, collected link utilization data was separatedinto two parts. The first part was used to construct the modelparameters used to forecast later data, which was the second part of thedata. The second part of the data was used to evaluate the forecast madeby the selected model. In accordance with the above, six months ofcollected link utilization data was used to evaluate the accuracy of theforecast made based upon the first portion of the link utilization data.Of course, in actual practice the methods in accordance with the presentinvention would be used to forecast future link utilization.

A known methodology referred to as the Box-Jenkins methodology may beused to fit linear time series models. This procedure requires the stepsof determining the number of differencing operations needed to render atime series stationary, determining the values of p and q, estimatingthe polynomials ω and θ, and evaluating how well the derived model fitsthe collected data. The estimation of the model parameters may be doneusing a maximum likelihood estimation. The model chosen as the bestmodel and used for forecasting purposes may be the one that provides thesmallest statistical fitting indices while also offering the smallestmean square prediction error six months ahead. Examples of acceptablestatistical fitting indices are forward predictive error (FPE), AkaikeInformation Criterion (AICC), and BIC.

The models computed for the long-term trend l(j) in the present exampleindicate that the first difference of these time series is consistentwith a simple moving average model with one or two terms plus a constantvalue μ. The need for one difference in operation at lag-1 and theexistence of the term μ across all models indicate that the long-termtrend across the traces for collected data is a simple exponentialsmoothing with growth. The trajectory of the long-term forecastresulting will typically be a sloping line with a slope equal to μ. Forexample, for trace-1 the long-term forecast corresponded to a weeklyincrease of 0.5633 megabits per second. This forecast represents theaverage aggregate demand of a link in the aggregate in the future. Theweekly increase in total demand between two adjacent Points of Presencecan thus be estimated through the multiplication of this value with thetotal number of active links in the aggregate between that pair of PoPs.Based upon this analysis and the estimates of μ across all models, itcan be observed that in the present example all traces exhibit upwardtrends, but at different rates of growth.

A similar process may be used to forecast future deviation of linkutilization by applying the Box-Jenkins methodology to the deviationmeasurements. It should be noted that in the present example some modelsof the deviation can be expressed with simple autoregressive models,while others can be accurately modeled as a moving average process afterone differencing operation. For example, the deviation for traces one,five, and eight increase with time at rates one order of magnitudesmaller than the increase in their long-term trends, while the deviationfor traces four and six can be approximated with a weighted movingaverage, which indicates a slower evolution.

From the above discussion, it can be seen that in the present exampleone cannot arrive at a single network wide forecasting model for thelink utilization between pairs of PoPs. Different parts of the IPnetwork grow at different rates, which is expressed as the long-termtrend forecast, and different parts of the IP network also experiencedifferent types of variation, which is the deviation around thelong-term trend.

The above models may be used to predict a baseline aggregate demand fora particular week in the future, and may forecast deviations around thatbaseline. The overall forecast for inter PoP aggregate demand may thenbe calculated using the prior stated:{circumflex over (x)}(j)=l(j)+3dt ₃(j)

While the forecast deviation may be above or below the forecast linkutilization, it should be noted that the deviation above the forecast islikely to be of greater importance for purposes of capacity planning, asa deviation below the forecast baseline would not require additionalnetwork resources.

Referring now to FIG. 15, collected link utilization data including thesecond part of the data and the forecast link utilization areillustrated for comparison. As can be seen in reference to FIG. 15, thebehavior forecast in accordance with the present invention based uponcollected link utilization data closely reflects actual collected linkutilization data for the forecast period.

To quantify the quality of the predictions with the observed linkutilization data, the multiresolution analysis may be applied to themeasurements in the evaluation period. The long-term trend l(j) may thenbe calculated, as may the weekly deviation dt₃(j) for each week in theforecast period. Using the above equation, {circumflex over (x)}(j) maythen be computed. Finally, the error in the derived forecast may becalculated as the forecast value minus {circumflex over (x)}(j), dividedby {circumflex over (x)}(j).

Referring now to FIG. 16, the relative error between the derivedforecast and {circumflex over (x)}(j) for each week in the evaluationperiod is illustrated. A negative error illustrated in FIG. 16 indicatesthat the actual demand was higher than the forecast demand. As shown inFIG. 16, the forecast error fluctuates with time but is centered nearzero. This means that on average aggregate demand was correctly forecastin this example. Twenty-four weeks into the future the forecastprediction error was 4%. The average prediction error across all weekswas −3.6%. For all five traces for which future link utilizationbehavior was forecast, the average absolute relative prediction errorwas lower than 15%.

The above-described invention is particularly useful for collecting,modeling, and forecasting traffic between adjacent PoPs in an IPnetwork. While the above methods may be used to forecast IP trafficvolume and deviation over many time frames, it is particularly wellsuited to the forecast of demand weeks or months in advance. As oneskilled in the art will realize, longer term forecasts, for examplethose exceeding six months, may be made using the present invention butwill be subject to greater uncertainty.

What is claimed is:
 1. One or more nontransitory computer-readable mediafor causing one or more computing devices to perform a method forforecasting the deviation of future link utilization between a pair ofpoints of presence in an IP network, the method comprising: collectingprior link utilization information, the prior link utilizationinformation identifying the aggregate link utilization between the pairof points of presence as a function of time; modeling the prior linkutilization using wavelet multiresolution analysis; identifying as adeviation approximation curve a curve from the wavelet multiresolutionanalysis that models the deviation of prior link utilization around thelong-term trend of the link utilization; constructing at least onelinear time series model of the deviation approximation curve; andforecasting future deviation of link utilization using one of the atleast one linear time series models of the deviation approximationcurve.
 2. The one or more nontransitory computer-readable media of claim1, wherein collecting prior link utilization information comprisescomputing average link utilization demand over ninety minute intervals.3. The one or more nontransitory computer-readable media of claim 2,wherein modeling the prior link utilization information using waveletmultiresolution analysis comprises: modeling the prior link utilizationinformation using an a-trous wavelet transform; and identifying as adeviation approximation curve a curve from the wavelet multiresolutionanalysis that models the deviation of prior link utilization around thelong-term trend of the link utilization comprises selecting the thirdtime scale a-trous detail signal as the deviation approximation curve.4. The one or more nontransitory computer-readable media of claim 3,wherein constructing at least one linear time series model of thedeviation approximation curve comprises: using the Box-Jenkinsmethodology to identify an ARIMA model that best fits the deviationapproximation curve.
 5. The one or more nontransitory computer-readablemedia of claim 4, wherein evaluating each of the at least one lineartime series models of the deviation approximation curve to determinewhich best matches the deviation of the prior link utilizationinformation comprises: determining that the ARIMA model of the deviationapproximation curve with the lowest AICC, BIC, and FPE measures bestmatches the deviation of the prior link utilization information.
 6. Oneor more nontransitory computer-readable media for causing one or morecomputing devices to perform a method for forecasting future linkutilization demand between a pair of points of presence in an IPnetwork, the method comprising: collecting prior link utilizationinformation, the prior link utilization information identifying theaggregate link utilization between the pair of points of presence as afunction of time; modeling the prior link utilization information usingwavelet multiresolution analysis to create an approximation curve thatmodels the long-term trend of the prior link utilization information anda deviation approximation curve that models the deviation of linkutilization around the long-term trend; constructing at least one lineartime series model of the approximation curve; constructing at least onelinear time series model of the deviation approximation curve;evaluating each of the at least one linear time series models of theapproximation curve to determine which best matches the long-term trendof the prior link utilization information; evaluating each of the atleast one linear time series models of the deviation approximation curveto determine which best matches the deviation around the long-term trendof the prior link utilization information; forecasting future linkutilization using the linear time series model of the approximationcurve that best matches the long-term trend of the prior linkutilization information; and forecasting future deviation of linkutilization using the linear time series model that best matches thedeviation of the prior link utilization.
 7. The one or morenontransitory computer-readable media of claim 6, wherein collectingprior link utilization information comprises computing aggregate linkutilization demand for ninety minute intervals.
 8. The one or morenontransitory computer-readable media of claim 7, wherein modeling theprior link utilization information using wavelet multiresolutionanalysis to create an approximation curve that models the long-termtrend of the prior utilization information comprises: modeling the priorlink utilization information using an a-trous wavelet transform; usingthe sixth time scale a-trous approximation as the approximation curve tomodel the long-term trend of the prior link utilization information; andusing the third time scale a-trous detail signal as the deviationapproximation curve that models the deviation of link utilization aroundthe long-term trend.
 9. The one or more nontransitory computer-readablemedia of claim 8, wherein constructing at least one linear time seriesmodel of the approximation curve comprises: using the Box-Jenkinsmethodology to identify an ARIMA model that best fits the long-termtrend of the prior link utilization data.
 10. The one or morenontransitory computer-readable media of claim 9, wherein evaluatingeach of the at least one linear time series models of the deviationapproximation curve to determine which best matches the deviation of theprior link utilization information comprises: using the Box-Jenkinsmethodology to identify an ARIMA model that best fits the deviation ofthe prior link utilization information.
 11. A method for forecastingfuture link utilization demand between a pair of points of presence inan IP network based upon prior link utilization information, the methodcomprising: modeling the prior link utilization using waveletmultiresolution analysis to create a plurality of curves that combine tosynthesize the prior link utilization information; using the analysis ofvariance technique to identify from the plurality of approximationcurves: A) a long-term trend approximation curve that best matches thelong-term trend of the prior link utilization information, and B) adeviation approximation curve that models the deviation around thelong-term trend of the prior link utilization information; constructingat least one linear time series model of the long-term approximationcurve; constructing at least one linear time series model of thedeviation approximation curve; selecting one of the at least one lineartime series model of the long-term trend approximation curve that bestmatches the prior link utilization information; selecting one of the atleast one linear time series model of the deviation approximation curvethat best matches the deviation around the long-term trend of the priorlink utilization information; forecasting future link utilization usingthe selected linear model of the long-term trend approximation curve;and forecasting future deviation of link utilization demand using theselected linear model of the deviation approximation curve.
 12. Themethod for forecasting future link utilization demand of claim 11,wherein selecting one of the at least one linear time series model ofthe long-term trend approximation curve comprises: using the Box-Jenkinsmethodology to fit an ARIMA model that best matches the long-term trendof the prior link utilization information.
 13. The method forforecasting future link utilization demand of claim 12, whereinselecting one of the at least one linear time series model of thedeviation approximation curve comprises: using the Box-Jenkinsmethodology to fit an ARIMA model that best matches the deviation of theprior link utilization information around the long-term trend.
 14. Themethod for forecasting future link utilization demand of claim 11,wherein modeling the prior link utilization information using waveletmultiresolution analysis comprises modeling the prior link utilizationinformation using an a-trous wavelet transform.